Experience First

This is always one of my favorite lessons to teach! Get ready for some big a-ha moments from your students. In this lesson we move from calculating instantaneous rate of change at various points to a  “slopes graph” that gives the instantaneous rate of change (termed “IROC” by one of our students) at any x-value. Students leave class today with a solid, tangible, well-conceptualized understanding of the derivative as a function, not just as individual slopes. The artifacts from this lesson are kept on our walls throughout the year and will be referred back to in many future lessons.

 

To prepare for today’s lesson, prepare two pieces of poster paper with a scaled graph. Label the y-axis “Slopes of f(x)” on the first poster and “Slopes of g(x)” on the second poster. You will need dot stickers as well.

 

On the front page, students look at a linear function and calculate the instantaneous rate of change at x=1 and x=3. In 1c, students generalize their findings and make a prediction about what the slopes graph would look like if we plotted the slope at any point. Since the function is linear, the slope at any x-value, even x=55, would be -2. Students see for the first time that the slopes graph of a linear function is a horizontal line at the value of the slope.

 

After debriefing the front page, assign each student a different number between -8 and 8 (a smaller or larger interval can be used depending on the size of your class). Students should record this value in the blank in 3a.

 

In question 2, groups will begin by describing the slopes of the quadratic parent function, specifically where the slopes are negative, positive, and zero.

 

In question 3, each student calculates the slope at an individual point using the limit definition, then marks their slope on the slopes graph with a dot sticker. The class graph should look like the line y=2x. Questions 3b and 3c challenge students to find the equation of the slopes graph. Remember, this is a pretty novel idea as of today--the fact that there is an equation that would “spit out” the slope at any x-value is pretty radical! Why is having an equation for the slopes so helpful? Because we can predict the slope at a different x-value (like x=55) without having to evaluate the limit of the difference quotient at a=55.

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Formalize Later

Engage in a series of noticings and wonderings about the slopes graph and the original function. Where does g(x) have positive slopes? How can we tell this from the slopes graph? Are the slopes getting steeper or flatter? Why does it make sense that the outputs of the slopes graph are getting higher as x increases? What happens to the slopes on the left side of the x-axis? How do we see this on the slopes graph?

 

After the dots/slopes are plotted, tell students the slopes graph has a special name. It is called a derivative and is denoted with f’(x) (pronounced “F prime of x”). Add this notation to your poster paper. Then point to a specific dot and ask which student is responsible for putting up that dot. Ask the student to explain what their dot represents. Do this multiple times during the lesson and in subsequent days so that students are familiarizing themselves with the idea that the point on the derivative represents the slope of the original curve at x=___”. Insist on hearing the word “slope”, the particular x-value, and that the slope is from the original function. It is very important that students don’t use vague language like “it” or “the graph” or even “the slope”. Train your students to use precise language like “the graph of f’” or “the slope of the original curve, f(x)”.

 

Ask students if they can come up with an equation for the derivative of g. If students are stuck, look at patterns in the ordered pairs. “At x=1, g has a slope of 2, at x=2, g has a slope of 4, at x=3, g has a slope of 6. What seems to be the relationship between the x-value and the slope at that x-value?”

 

We have chosen to use Newtonian notation (prime notation) to refer to derivatives instead of Leibnitz notation (dy/dx). In Precalculus, the emphasis is on building the conceptual idea of the derivative and we find that using one consistent notation aids in this. In AP Calculus we add the Leibnitz notation to the mix.